Computational Statistics in Data Science - Группа авторов

Скачать книгу

ModifyingAbove normal upper Lamda With Ì‚ Subscript i i comma n Baseline equals StartFraction 1 Over n EndFraction sigma-summation Underscript t equals 1 Overscript n Endscripts left-parenthesis h Subscript i Baseline left-parenthesis upper X Subscript t Baseline right-parenthesis minus ModifyingAbove theta With Ì‚ Subscript i comma h Baseline right-parenthesis squared equals StartFraction 1 Over n EndFraction sigma-summation Underscript t equals 1 Overscript n Endscripts left-bracket h Subscript i Baseline left-parenthesis upper X Subscript t Baseline right-parenthesis right-bracket squared minus left-bracket ModifyingAbove theta With Ì‚ Subscript i comma h Baseline right-bracket squared

      We obtain the asymptotic distribution of ModifyingAbove normal upper Lamda With Ì‚ Subscript i i comma n. A similar argument can be made for the off‐diagonals of normal upper Lamda. Under the conditions of Theorem 1,

StartRoot n EndRoot left-parenthesis StartBinomialOrMatrix n Superscript negative 1 Baseline sigma-summation h Subscript i Baseline left-parenthesis upper X Subscript t Baseline right-parenthesis squared Choose ModifyingAbove theta With Ì‚ Subscript i comma h Baseline EndBinomialOrMatrix minus StartBinomialOrMatrix normal upper E Subscript upper F Baseline left-bracket h Subscript i Superscript 2 Baseline right-bracket Choose theta Subscript i comma h Baseline EndBinomialOrMatrix right-parenthesis right-arrow Overscript d Endscripts upper N 2 left-parenthesis 0 comma sigma-summation Underscript normal upper Lamda Subscript i i Baseline Endscripts right-parenthesis

      where upper Sigma Subscript normal upper Lamda Sub Subscript i i is

upper Sigma Subscript normal upper Lamda Sub Subscript i i Baseline equals sigma-summation Underscript k equals negative infinity Overscript infinity Endscripts Start 2 By 2 Matrix 1st Row 1st Column Cov Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis squared comma h Subscript i Baseline left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis squared right-parenthesis 2nd Column Cov Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis squared comma h Subscript i Baseline left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis right-parenthesis 2nd Row 1st Column left-bracket Cov Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis squared comma h Subscript i Baseline left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis right-parenthesis right-bracket Superscript upper T Baseline 2nd Column Cov Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis comma h Subscript i Baseline left-parenthesis upper X Subscript 1 plus k Baseline right-parenthesis right-parenthesis EndMatrix

      Under IID sampling, the infinite sum above reduces to

upper Sigma Subscript normal upper Lamda Sub Subscript i i Superscript IID Baseline equals Start 2 By 2 Matrix 1st Row 1st Column upper V a r Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis squared right-parenthesis 2nd Column Cov Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis squared comma h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis right-parenthesis 2nd Row 1st Column left-bracket Cov Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis squared comma h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis right-parenthesis right-bracket Superscript upper T Baseline 2nd Column upper V a r Subscript upper F Baseline left-parenthesis h Subscript i Baseline left-parenthesis upper X 1 right-parenthesis right-parenthesis EndMatrix

      Applying the delta method for function phi left-parenthesis x comma y right-parenthesis equals x minus y squared, we obtain

StartRoot n EndRoot left-parenthesis ModifyingAbove normal upper Lamda With Ì‚ Subscript i i comma n Baseline minus normal upper Lamda Subscript i i Baseline right-parenthesis right-arrow Overscript d Endscripts upper N left-parenthesis 0 comma Start 1 By 2 Matrix 1st Row 1st Column 1 2nd Column minus 2 mu Subscript h Baseline EndMatrix upper Sigma Subscript normal upper Lamda Sub Subscript i i Subscript Baseline StartBinomialOrMatrix 1 Choose minus 2 mu Subscript h Baseline EndBinomialOrMatrix right-parenthesis

      3.4 Confidence Regions for Means

      Suppose that upper A Subscript n is an estimate of the limiting Monte Carlo variance–covariance matrix, normal upper Lamda for IID sampling, and upper Sigma for MCMC sampling. Let chi Subscript 1 minus alpha comma p Superscript 2 be the left-parenthesis 1 minus alpha right-parenthesis‐quantile of a chi Subscript p Superscript 2 distribution. The CLT yields a large‐sample confidence region around ModifyingAbove theta With Ì‚ Subscript h as

upper C Subscript alpha Superscript upper E Baseline left-parenthesis ModifyingAbove theta With Ì‚ Subscript h Baseline right-parenthesis equals StartSet theta element-of double-struck upper R Superscript p Baseline colon n left-parenthesis theta Subscript n Baseline minus theta right-parenthesis Superscript upper T Baseline upper A Subscript n Superscript negative 1 Baseline left-parenthesis theta Subscript n Baseline minus theta right-parenthesis less-than chi Subscript 1 minus alpha comma p Superscript 2 Baseline EndSet

      Let StartAbsoluteValue dot EndAbsoluteValue denote the determinant. The volume of this ellipsoidal confidence region, which depends on p, alpha, and StartAbsoluteValue upper A Subscript n Baseline EndAbsoluteValue, is given by